On Extracting Maximum Stable Sets in Perfect Graphs Using Lovász's Theta Function

We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovász's theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its induced subgraphs. We develop an efficient, polynomial-time algorithm to extract a maximum stable set in a perfect graph using the theta function. Our algorithm iteratively transforms an approximate solution of the semidefinite formulation of the theta function into an approximate solution of another formulation, which is then used to identify a vertex that belongs to a maximum stable set. The subgraph induced by that vertex and its neighbors is removed and the same procedure is repeated on successively smaller graphs. We establish that solving the theta problem up to an adaptively chosen, fairly rough accuracy suffices in order for the algorithm to work properly. Furthermore, our algorithm successfully employs a warm-start strategy to recompute the theta function on smaller subgraphs. Computational results demonstrate that our algorithm can efficiently extract maximum stable sets in comparable time it takes to solve the theta problem on the original graph to optimality. This work was supported in part by NSF through CAREER Grant DMI-0237415. Part of this work was performed while the first author was at the Department of Applied Mathematics and Statisticsat Stony Brook University, Stony Brook, NY, USA.     

Microcrystals formed in proton bombarded poly(vinyl chloride) films

Thin films of PVC were bombarded with 700 keV protons, and studied with RGA during bombardment, and later with TEM. A great mass loss was observed, and the remaining material was found to contain many micro-crystals with interplanar spacings of 0.25 and 0.28 nm within an amorphous matrix.     

New glyoxime derivatives and their transition metal complexes

Two new vic-dioxime ligands and their complexes with Co⁺², Ni⁺², Cu⁺², Cd⁺², and Zn⁺² ions were synthesized. Primer amines (3,4-methylenedioxaaniline and 4-methylbenzylamine) reacted with antichloroglyoxime to give 3,4-methylenedioxaphenylaminoglyoxime (H₂L¹) and N-(4-methylbenzyl)aminoglyoxime (H₂L²) ligands. Structures of the ligands and their complexes are proposed based on elemental analyses, IR, UV-Vis, and ¹H NMR spectra, magnetic susceptibility measurements, and thermogravimetric analyses (TGA). The article was submitted by the authors in English.     

A note on

There is only one pair of non-real zeros of , and of , in the left half-plane. The Riemann Hypothesis implies that and have no zeros in the strip .

     

Anomalous NMR behavior of meso compounds with remote stereogenic centers on addition of chiral shift reagent or chiral solvating agent

A problem has arisen in using chiral shift reagents (CSR) and chiral solvating agents (CSA) to determine meso and racemic forms of diastereoisomers in which the stereogenic centers of the molecules are separated by achiral spacers. It is found that NMR signals of both meso and racemic forms of diastereoisomers may exhibit doubling on addition of CSR/CSA, which means that unequivocal assignments cannot be made without characterizing the effects for separate meso and racemic forms; this is particularly important for additions of CSR/CSA at relatively low concentrations, which always result in the splitting of some NMR signals of diastereoisomers. The phenomenon is demonstrated in the (31)P NMR spectra of meso and racemic forms of three spermine-bridged gem-disubstituted cyclotriphosphazatrienes, 1a-c, and compared with analogous achiral molecules, the per-substituted spermine-bridged cyclotriphosphazatrienes 2a-d. As expected, only one set of (31)P NMR signals was observed for the achiral compounds 2a-d, even on addition of CSA. Two sets of (31)P NMR ABX multiplets corresponding to meso and racemic diastereoisomers were observed for compounds 1a-c; on addition of CSA, the signals of at least one of the multiplets for each compound separated into more than the expected groups of three lines with an intensity distribution of 2:1:1. To understand this phenomenon, the meso and racemic forms of 1a and 1b and the meso form of 1c have been separated and characterized by X-ray crystallography. On addition of CSA to the racemic forms of 1a and 1b, the (31)P NMR spectrum shows the expected doubling of signals, but, unexpectedly, the same is observed for each of the meso forms of 1a-c. Analogous results using both CSA and CSR have been obtained for the meso and racemic forms of the diastereoisomeric piperazine-bridged macrocyclic-phosphazene compound, 3, whereas no effect was observed for the two meso forms of the doubly bridged macrocyclic-phosphazene compound 4. The phenomenon of doubling of the (31)P NMR signals of the meso form of singly bridged cyclotriphosphazatrienes, 1a-c and 3, is explained by consideration of the equilibrium in solution of independent complexation of a chiral ligand with molecules that have two chiral cyclophosphazene moieties separated by an achiral spacer group. The results show that the stereogenicity of such diastereoisomeric molecules in solution cannot be characterized unequivocally by NMR measurements on addition of either CSR or CSA.     

Poly(2-(dimethylamino)ethyl methacrylate) brushes fabricated by surface-mediated RAFT polymerization and their response to pH

In this study, well-defined, high density poly(2-(dimethylamino)ethyl methacrylate) [poly(DMAEMA)] brushes were fabricated by the combination of the self-assembly of a monolayer of RAFT agent and surface-mediated RAFT polymerization. The whole fabrication process of the poly(DMAEMA) was followed by water contact angles, grazing angle-Fourier transform infrared spectroscopy, X-ray photoelectron spectroscopy, and atomic force microscopy. Kinetic studies revealed a linear increase in poly(DMAEMA) film thickness with polymerization time, indicating that the chain growth from the surface was a controlled process. Characterization of the poly(DMAEMA) brushes, such as molecular weight and thickness determination, were measured by gel permeation chromatography, and ellipsometry, and the grafting density was estimated. The pH response of the poly(DMAEMA) brushes was further investigated and the results verified the “brush-like” to “mushroom-like” transition of the poly(DMAEMA) chains due to the reversible protonation/deprotonation upon changing the solution pH.     

Bioadhesion to solids: contact angle hysteresis effect

It was recently reported that the ease of removal of sporelings of green seaweed Ulva under shear stress from the polymer surfaces was found to be linearly and positively correlated with contact angle and wetting hysteresis, i.e., the higher the hysteresis, the greater the removal. Motivated by this report, we examined the relationship between the bioadhesion of blood platelets and proteins with contact angle hysteresis of solid substrates using the data of published papers. It was determined that there is a linear and positive relationship between the contact angle hysteresis and bioadhesion of both blood platelets and γ-globulin protein contacting the solid substrates, i.e., the higher the hysteresis, the greater the bioadhesion. The reasons are discussed and it is proposed that testing the effect of CAH on the adhesion strengths of biomaterials on surfaces is useful in order to gain a better insight on the bioadhesion mechanism.     

A practical access to new pyrrolizine carboxylates via KHMDS-catalyzed carbocyclizations

Easy and effective preparation of new 1H-pyrrolizine carboxylates was achieved with high efficiency via KHMDS-induced carbocyclization of N-alkynyl proline carboxylates under substantially mild conditions. Meanwhile, some trans-diiodoallylic N-proline carboxylates were obtained from N-propargyl proline carboxylates using molecular I₂ with or without KHMDS. This method is quite feasible in terms of practical and quick access to the pyrrolizines and their derivatives over the formation of carbanions.